62 



C. V. L. Charlier 



be supposed the form of Maxwell. Tliis method has the advantage that even the 

 inverse problem thus can be solved. From the observations the form of '^{m,u,v) 

 is, indeed, known. Combining (2), (30) and (31) we find now the formula (12) 

 (p. 4 of this memoir) 



v'.. = N'.., 



where ■O-« denotes the mean value of 1 : r*. 



Through the help of this formula the relations (19) and the following ones 

 have been derived which allow the calculation of the moments Ny (about the mean) 

 of the unknown function (w, Ü, V). 



The essential difference in the appearance of the frequency curve of the ap- 

 parent proper motions and that of the linear velocities arises from the properties 

 of the function O-g. We have indeed, according to (16), 



(16) ■ ^^=p'q'\ 



where 



(32) q=^e 



If q were equal to unity, then it follows from the formulée (20) and the follow- 

 ing ones — which can be generally proved — that the moments Nij should be 

 proportional to Vy. In such a case the frequency curve of the proper motions should 

 really be of the same form as that of the linear velocities. The function g can, 

 however, be equal to unity only for \^ — 1 (the value = 0 leads indeed to no 

 solution) h. e. only for = 0. 



In other words: the frequency curve of the proper motions should be equal 

 to (or proportional to) that of the linear velocities only if all stars should have the 

 same absolute magnitude. The law of the densities — I){r) — might then be of any 

 form. We always get a proportionality between these two frequency distributions. 



Now such a value as = 1 is certainly excluded. The absolute magnitude 

 of the stars is far from being constant. Without doubt the value of Xj is, still, 

 rather unperfectly known. I shall discuss beneath the value found from the proper 

 motions of the brighter stars. 



On account of the fundamental importance of this quantity I have calculated 

 the moments Ny under two different assumptions for q — properly chosen for 

 taking into regard the values that here may be taken into consideration. It is 

 however found — which could not, indeed, be a priori anticipated — that this un- 

 certainty in tiie value of q has a very small influence of the resulting appearance 

 of the frequency curve of the hnear velocities. 



It might be objected that this conclusion is based on the results found in I 

 regarding the density function and regarding the frequency function of the absolute 

 magnitude. It is however easy to prove that similar conclusions must hold true 

 for any acceptable hypothesis regarding these functions. Thus the hypothesis of 

 Seeliger (I (12)) lead to the form (16) for i% and even the simple assumption 

 D — Gonstans gives a similar result. 



